cp's OEIS Frontend

A073004 Decimal expansion of exp(gamma).

%I A073004 #95 Aug 28 2025 00:32:55
%S A073004 1,7,8,1,0,7,2,4,1,7,9,9,0,1,9,7,9,8,5,2,3,6,5,0,4,1,0,3,1,0,7,1,7,9,
%T A073004 5,4,9,1,6,9,6,4,5,2,1,4,3,0,3,4,3,0,2,0,5,3,5,7,6,6,5,8,7,6,5,1,2,8,
%U A073004 4,1,0,7,6,8,1,3,5,8,8,2,9,3,7,0,7,5,7,4,2,1,6,4,8,8,4,1,8,2,8,0,3,3,4,8,2
%N A073004 Decimal expansion of exp(gamma).
%C A073004 See references and additional links in A094644.
%C A073004 The Riemann hypothesis holds if and only if the inequality sigma(n)/(n*log(log(n))) < exp(gamma) is valid for all n >= 5041, (G. Robin, 1984). - _Peter Luschny_, Oct 18 2020
%C A073004 From _Peter Bala_, Aug 24 2025: (Start)
%C A073004 By definition, gamma = lim_{n -> oo} s(n), where s(n) = Sum_{k = 1..n} 1/k - log(n). The convergence is slow. For example, s(50) = 0.5(87...) is only correct to 1 decimal digit. Let S(n) = Sum_{k = 0..n} (-1)^(n+k)*binomial(n, k)*binomial(n+k, k)*s(n+k). Elsner shows that S(n) converges to gamma much more rapidly. For example, S(50) = 0.57721566490153286060651209008(02...) gives gamma correct to 29 decimal digits.
%C A073004 Define E(n) = Sum_{k = 0..n} (-1)^(n+k)*binomial(n, k)*binomial(n+k, k)*exp(s(n+k)). Then it appears that E(n) converges rapidly to exp(gamma). For example, E(50) = 1.78107241799019798523650410310(43...) gives exp(gamma) correct to 29 decimal digits. Cf. A002389. (End)
%D A073004 Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Sections 1.5.1 and 2.27.2, pp. 31, 187.
%D A073004 Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See pp. 166, 191, 208.
%H A073004 Stanislav Sykora, <a href="/A073004/b073004.txt">Table of n, a(n) for n = 1..2000</a>
%H A073004 C. Elsner, <a href="https://doi.org/10.1090/S0002-9939-1995-1233969-4">On a sequence transformation with integral coefficients for Euler's constant</a>, Proc. Amer. Math. Soc., Vol. 123 (1995), Number 5, pp. 1537-1541.
%H A073004 Paul Erdős and S. K. Zaremba, <a href="https://users.renyi.hu/~p_erdos/1974-34.pdf">The arithmetic function Sum_{d|n} log d/d</a>, Demonstratio Mathematica, Vol. 6 (1973), pp. 575-579.
%H A073004 T. H. Gronwall, <a href="https://www.jstor.org/stable/1988773">Some Asymptotic Expressions in the Theory of Numbers</a>, Trans. Amer. Math. Soc., Vol. 14, No. 1 (1913), pp. 113-122.
%H A073004 Jeffrey C. Lagarias, <a href="http://arxiv.org/abs/1303.1856">Euler's constant: Euler's work and modern developments</a>, arXiv:1303.1856 [math.NT], 2013.
%H A073004 Jeffrey C. Lagarias, <a href="https://doi.org/10.1090/S0273-0979-2013-01423-X">Euler's constant: Euler's work and modern developments</a>, Bull. Amer. Math. Soc., 50 (2013), 527-628.
%H A073004 Simon Plouffe, <a href="http://www.worldwideschool.org/library/books/sci/math/MiscellaneousMathematicalConstants/chap24.html">The exp(gamma)</a>.
%H A073004 G. Robin, <a href="http://zakuski.utsa.edu/~jagy/Robin_1984.pdf">Grandes valeurs de la fonction somme des diviseurs et hypothèse de Riemann</a>, J. Math. Pures Appl. 63 (1984), 187-213.
%H A073004 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Euler-MascheroniConstant.html">Euler-Mascheroni Constant</a>.
%H A073004 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GronwallsTheorem.html">Gronwall's Theorem</a>.
%H A073004 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/MertensTheorem.html">Mertens Theorem</a>, Equations 2-3.
%H A073004 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RobinsTheorem.html">Robin's Theorem</a>.
%F A073004 By Mertens theorem, equals lim_{m->infinity}(1/log(prime(m))*Product_{k=1..m} 1/(1-1/prime(k))). - _Stanislav Sykora_, Nov 14 2014
%F A073004 Equals limsup_{n->oo} sigma(n)/(n*log(log(n))) (Gronwall, 1913). - _Amiram Eldar_, Nov 07 2020
%F A073004 Equals limsup_{n->oo} (Sum_{d|n} log(d)/d)/(log(log(n)))^2 (Erdős and Zaremba, 1973). - _Amiram Eldar_, Mar 03 2021
%F A073004 Equals Product_{k>=1} (1-1/(k+1))*exp(1/k). - _Amiram Eldar_, Mar 20 2022
%F A073004 Equals lim_{n->oo} n * Product_{prime p<=n} p^(1/(1-p)). - _Thomas Ordowski_, Jan 30 2023
%F A073004 Equals Product_{k>=1} (k/sqrt(2))^((-1)^k/(k*log(2))). - _Antonio Graciá Llorente_, Oct 11 2024
%F A073004 Equals lim_{n->oo} (1/log(n))*Product_{prime p<=n} p/(p - 1) [Mertens] (see Finch at p. 31). - _Stefano Spezia_, Oct 27 2024
%e A073004 Exp(gamma) = 1.7810724179901979852365041031071795491696452143034302053...
%t A073004 RealDigits[ E^(EulerGamma), 10, 110] [[1]]
%o A073004 (PARI) exp(Euler)
%o A073004 (Magma) R:=RealField(100); Exp(EulerGamma(R)); // _G. C. Greubel_, Aug 27 2018
%Y A073004 Cf. A001620 (Euler-Mascheroni constant, gamma).
%Y A073004 Cf. A001113, A002389, A067698, A080130, A091901, A094644 (continued fraction for exp(gamma)), A155969, A246499.
%K A073004 cons,nonn,easy,changed
%O A073004 1,2
%A A073004 _Robert G. Wilson v_, Aug 03 2002